The Parallelogram Identity

Idea

The parallelogram identity is an identity which characterises those norms which are the norms associated with inner products. An inner product can be considered as being the structure required to define the angle between two vectors and a norm can be considered as being the structure required to define the length of a vector. From standard Euclidean geometry, lengths and angles (almost) determine each other so knowing one, we should be able to define the other.

If length is known, angle can be defined by the cosine law:

c2=a2+b2−2abcosC
c^2 = a^2 + b^2 - 2 a b \cos C

Although this formula could be used to define “angle” for any length (that is, norm), not every length admits a sensible notion of an angle. There are several ways to describe what the fundamental properties of angles should be. One is to say that angles should add:

θ(u,v)+θ(v,w)=θ(u,w)
\theta(u,v) + \theta(v,w) = \theta(u,w)

This needs careful interpretation since the angle between two vectors is slightly ambiguous: one can choose the internal angle or the external angle and it is not possible to make a consistent choice. However, modulo that uncertainty, the above is a reasonable property to insist on.

A special case of this is when w=−uw = -u. This leads to the following diagram.

By imposing the assumption that angles should add correctly, we deduce that ψ=π−ϕ\psi = \pi - \phi and thus cos(ψ)=−cos(ϕ)\cos (\psi) = - \cos(\phi). Summing the two lines above leads to the parallelogram identity:

If a norm satisfies this identity, then the definition of angle (using the cosine identity) satisfies all the basic properties of angles that one can consider.

Inner Products

A norm which satisfies the parallelogram identity is the norm associated with an inner product. Using the parallelogram identity, there are three commonly stated equivalent forumlae for the inner product; these are called the polarization identities. (The notation can vary a little, but it is usually some form of round or angled brackets.) These formulae hold for vector spaces over ℝ\mathbb{R}; there are similar formulae for ℂ\mathbb{C}, but they have more terms.

To get it as stated, we then apply this in the special case of w=0w = 0 to deduce that ⟨u,v⟩=2⟨12u,v⟩\langle u, v \rangle = 2\langle \frac{1}{2} u, v\rangle. Substituting this back in to the above, we obtain the form in the statement.

The properties above were all reasonably straightforward deductions from the definition. There is one more property that is needed which is a little more complicated. First, we note a useful result about the continuity of the supposed inner product.

Let WW be the linear subspace spanned by the image f(V)f(V). Then WW inherits an inner product, and if {ei}\{e_i\} is an orthonormal basis of VV (without restriction on cardinality), then {ui=f(ei)}\{u_i = f(e_i)\} is also an orthonormal basis of WW, since ff preserves the inner product. Next, for any linear combination ax+bya x + b y,

for each uiu_i, whence f(ax+by)=af(x)+bf(y)f(a x + b y) = a f(x) + b f(y). Thus ff is a linear map, and a linear isomorphism onto the subspace WW because it carries a basis eie_i of VV to a basis uiu_i of WW.